Many remote sensing instruments, (i.e. radars, lidars, sodars), calculate Doppler spectra (power spectra scaled to represent the Doppler shift) as part of their signal processing. Transforming signals from a time-series representation to a power spectrum is very useful since it concentrates the signal energy into a small region of the power spectrum while spreading out the noise energy. Typically, the desired signal appears as a peak in the power spectrum in the presence of noise. If the signal-to-noise ratio (SNR) is large enough, it is easy to detect the signal peak amongst the noise peaks. Usually, the largest peak in the spectrum is chosen as the signal. However, when the SNR is low, it is difficult to select the desired signal since the noise peaks can be larger than the signal. Optimum detectability occurs when the signal exists at least as long as the duration of the sampling time. However, many real-world signals are of shorter duration. These transient signals are difficult to detect in the presence of noise. Under these conditions, the signal energy is contained in many spectral points and difficult to detect.
Most instrument designers will choose the sampling time of an instrument equal to the time the scattering target is in the beam to optimize its detectability. Since the Fourier transform of a continuous signal has narrow bandwidth, its corresponding signal will have a narrow peak in the power spectrum. This narrow peak has the best chance of detection. In addition, the sampled time series is often divided into short blocks whose power spectra are averaged together. This averaging reduces the variance of the noise making signal peaks more detectable. However, often the detection of a transient signal is required. The Fourier transform of a transient signal will have a broad peak in the power spectrum. Therefore, the signal energy will appear in many frequency bins and be difficult to detect. Also, it is not possible to average short power spectra together when trying to detect transient signals, since averaging would reduce the amplitude of a transient peak.
Detecting signals in the presence of noise is a problem when signal amplitude is small compared with the amplitude of the noise signal. Transforming from a time-series representation to a power spectrum is very useful since it concentrates the signal energy into a small region of the power spectrum while spreading out the noise energy. Typically, the amplitude of a signal peak must be larger than one to two standard deviations of the noise peaks to be detected. The Heisenberg-Gabor Uncertainty Principal states that long duration time signals have narrow spectral peaks and short transient signals have broad spectral peaks. This makes detection of long duration signals much easier, since the signal energy will be contained in very few spectral bins. Transient signals with short time duration will spread the signal energy over a larger part of the spectrum, making detection based on the peak exceeding the noise deviations difficult.
This effect is demonstrated in FIGS. 1A-1C. FIG. 1A is a simulated power spectrum calculated from a 500 msec long time series. This spectrum contains exponentially distributed noise and a 400 msec long burst of a sine wave. The signal peak is contained in one spectral bin and is obviously detectable compared with the surrounding noise peaks. FIG. 1B is a similar spectrum that contains a 40 msec burst of signal with the same signal energy as in FIG. 1A. Again the signal peak is easily detectable, standing well above the noise. However, the signal peak has been spread over several frequency bins and is not as tall in relation to the noise peaks. The shorter duration signal has a larger spectral width and is not as detectable as the longer duration signal. FIG. 1C shows another spectrum that contains a 4 msec signal burst with the same signal energy as above. The peak is so broad that it is not possible to detect the signal in the noise. The solid line is added to FIG. 1C to highlight the signal peak. The spectral width of a 4 msec burst of signal is so large that it is not possible to choose the signal peak from the noise peaks. A different detection method is necessary to detect such short transient signals.